Posterior analysis for normalized random measures with independent increments

Scandinavian Journal of Statistics

Volumn 36, Issue 1, 2009, Pages 76-97

  James, Lancelot F ^a,b   Lijoi, Antonio ^c   PrÜnster, Igor ^a  

^a UNIVERSITY OF TURIN   (Italy)

^b HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Hong Kong)

^c UNIVERSITY OF PAVIA   (Italy)

Author keywords

Bayesian Nonparametrics; Dirichlet process; Normalized random measure; Poisson random measure; Posterior distribution; Predictive distribution

Indexed keywords

EID: 60249093564     PISSN: 03036898     EISSN: 14679469     Source Type: Journal    
DOI: 10.1111/j.1467-9469.2008.00609.x     Document Type: Article

References (48)

1. Antoniak, C. E. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist. 2, 1152 1174.
2. Crawford, S. L. (1994). An application of the Laplace method to finite mixture distributions. J. Amer. Statist. Assoc. 89, 259 267.
3. Daley, D. J. Vere-Jones, D. (1988). An introduction to the theory of point processes. Springer, New York.
4. Doksum, K. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2, 183 201.
5. Dykstra, R. L. Laud, P. W. (1981). A Bayesian nonparametric approach to reliability. Ann. Statist. 9, 356 367.
6. Escobar, M. D. West, M. (1995). Bayesian density estimation and inference using mixtures. J. Amer. Statist. Assoc. 90, 577 588.
7. Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theor. Popul. Biol. 3, 87 112.
8. Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1, 209 230.
9. Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2, 615 629.
10. Ferguson, T. S. Klass, M. J. (1972). A representation of independent increment processes without Gaussian components. Ann. Statist. 43, 1634 1643.
11. Griffin, J 2007). The Ornstein-Uhlenbeck Dirichlet process and other time-varying processes for Bayesian nonparametric inference. CRiSM Working Paper07-03. University of Warwick.
12. Hjort, N. L. (1990). Nonparametric Bayes estimators based on beta processes in models for life history data. Ann. Statist. 18, 1259 1294.
13. Ishwaran, H. James, L. F. (2001). Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc. 96, 161 173.
14. Ishwaran, H. James, L. F. (2003). Generalized weighted Chinese restaurant processes for species sampling mixture models. Statist. Sinica 13, 1211 1235.
15. James, L. F. (2002). Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics. Unpublished manuscript. MathArXiv math.PR/0205093.
16. James, L. F. (2003). A simple proof of the almost sure discreteness of a class of random measures. Statist. Probab. Lett. 65, 363 368.
17. James, L. F. (2005a). Poisson process partition calculus with applications to Bayesian Lévy moving averages and shot-noise processes. Ann. Statist. 33, 1771 1799.
18. James, L. F. (2005b). Functionals of Dirichlet processes, the Cifarelli-Regazzini identity and Beta-Gamma processes. Ann. Statist. 33, 647 660.
19. James, L. F. (2006). Poisson calculus for spatial neutral to the right processes. Ann. Statist. 34, 416 440.
20. James, L. F., Lijoi, A. Prünster, I. (2006). Conjugacy as a distinctive feature of the Dirichlet process. Scand. J. Statist. 33, 105 120.
21. Kallenberg, O. (1986). Random measures. Akademie Verlag, Berlin.
22. Kim, Y. (1999). Nonparametric Bayesian estimators for counting processes. Ann. Statist. 27, 562 588.
23. Kingman, J. F. C. (1967). Completely random measures. Pacific J. Math. 21, 59 78.
24. Kingman, J. F. C. (1975). Random discrete distributions (with discussion). J. Roy. Statist. Soc. Ser. B 37, 1 22.
25. Kingman, J. F. C. (1993). Poisson processes. Oxford University Press, Oxford.
26. Korwar, R. M. Hollander, M. (1973). Contributions to the theory of Dirichlet processes. Ann. Probab. 1, 705 711.
27. Lijoi, A., Mena, R. H. Prünster, I. (2005). Hierarchical mixture modelling with normalized inverse Gaussian priors. J. Amer. Statist. Assoc. 100, 1278 1291.
28. Lijoi, A., Mena, R. H. Prünster, I. (2007a). Controlling the reinforcement in Bayesian nonparametric mixture models. J. Roy. Statist. Soc. Ser. B Stat. Methodol. 69, 715 740.
29. Lijoi, A., Mena, R. H. Prünster, I. (2007b). Bayesian nonparametric estimation of the probability of discovering a new species. Biometrika 94, 769 786.
30. Lo, A. Y. (1982). Bayesian nonparametric statistical inference for Poisson point processes. Z. Wahrsch. Verw. Gebiete 59, 55 66.
31. Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates: I. Density estimates. Ann. Statist. 12, 351 357.
32. MacEachern, S. N. (1994). Estimating normal means with a conjugate style Dirichlet process prior. Commun. Statist. Simulation Comp. 23, 727 741.
33. MacEachern, S. N 1998). Computational methods for mixture of Dirichlet process models. In Practical nonparametric and semiparametric Bayesian statistics (eds D. Dey, P. Müller D. Sinha 23 43. Springer, New York.
34. McGrory, C. A., Titterington, D. M. (2007). Variational approximations in Bayesian model selection for finite mixture distributions. Comput. Statist. Data Anal. 51, 5352 5367.
35. Müller, P. Quintana, F. A. (2004). Nonparametric Bayesian data analysis. Statist. Sci. 19, 95 110.
36. Nieto-Barajas, L. E. Prünster, I. (2008). A sensitivity analysis for Bayesian nonparametric density estimators. Statist. Sinica (in press).
37. Papaspiliopoulos, O. Roberts, G. O 2008). Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models. Biometrika 95, 169 186.
38. Perman, M., Pitman, J. Yor, M. (1992). Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92, 21 39.
39. Pitman, J 1996). Some developments of the Blackwell-MacQueen urn scheme. In Statistics, Probability and Game Theory. Papers in honor of David Blackwell (eds T. S. Ferguson, L. S. Shapley J. B. MacQueen 245 267. Lecture Notes, Monograph Series, 30, IMS, Hayward.
40. Pitman, J 2003). Poisson-Kingman partitions. In Science and Statistics: A Festschrift for Terry Speed (ed. D. R. Goldstein 1 35. Lecture Notes, Monograph Series, 40, IMS, Hayward.
41. Pitman, J 2006). Combinatorial stochastic processes. Ecole d'Eté de Probabilités de Saint-Flour XXXII. Lecture Notes in Mathematics 1875. Springer, Berlin.
42. Pitman, J. Yor, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25, 855 900.
43. Prünster, I 2002). Random probability measures derived from increasing additive processes and their application to Bayesian statistics. Ph.d dissertation, University of Pavia.
44. Regazzini, E., Lijoi, A. Prünster, I. (2003). Distributional results for means of random measures with independent increments. Ann. Statist. 31, 560 585.
45. Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statist. Sinica 4, 639 650.
46. Walker, S. G. Damien, P. (2000). Representations of Lévy processes without Gaussian components. Biometrika 87, 477 483.
47. Walker, S. Muliere, P. (1997). Beta-Stacy processes and a generalization of the Pólya-urn scheme. Ann. Statist. 25, 1762 1780.
48. Wolpert, R. Ickstadt, K. (1998). Poisson/gamma random field models for spatial statistics. Biometrika 85, 251 267.